A Stationary Composition Law for Schwinger-Keldysh Effective Actions

Let $W_{\mathcal{O}}[J]$ and $W_{\mathcal{C}}[J]$ denote the connected generating functionals of an open system and its closed system counterpart, and define the environment-induced contribution $W_{\mathrm{I}\mathrm{F}}[J]$ by the difference: $W_{\mathrm{I}\mathrm{F}}[J]\equiv W_{\mathcal{O}}[J]-W_{\mathcal{C}}[J]\mathrm{.}$ The corresponding Legendre transforms, ${\mathrm{\Gamma }}_{\mathcal{O}}$, ${\mathrm{\Gamma }}_{\mathcal{C}}$, and ${\mathrm{\Gamma }}_{\mathrm{I}\mathrm{F}}$, do not obey an analogous additive relation but satisfy a stationary composition law given by: $${\mathrm{\Gamma }}_{\mathcal{O}}[\mathrm{\Phi }]=\underset{\mathrm{\Psi }}{\mathrm{Stat}}\{{\mathrm{\Gamma }}_{\mathcal{C}}[\mathrm{\Psi }]+{\mathrm{\Gamma }}_{\mathrm{I}\mathrm{F}}[\mathrm{\Phi }-\mathrm{\Psi }]\}$$ where $\mathrm{S}\mathrm{t}\mathrm{a}\mathrm{t}$ denotes evaluation at a solution of the stationarity condition with respect to $\mathrm{\Psi }$. This variational composition applies to both local and nonlocal Schwinger-Keldysh effective actions in nonequilibrium quantum field theory. We illustrate its linear and nonlinear realizations through quadratic theories and general time independent effective actions, respectively.